Известия
HAH Армении.
Математика,
том 44, и. 4, 2009, стр.
35-52.
APPROXIMATION OF FUNCTIONS BY POLYNOMIALS WITH
VARIOUS CONSTRAINTS
R. M. T R I G U B
E-mail:
[email protected]
АННОТАЦИЯ. The pointwise approximations of functions with a given modulus
of continuity of their r-th derivatives on a closed interval of the real axis are
done with various additional constraints. The paper attempts to review such
constraints.
1. I N T R O D U C T I O N
As a rule, pointwise approximations of functions with a given modulus of continuity
of their r-th derivatives on a closed interval of the real axis are done with various
additional constraints, such as:
A. Simultaneous approximation of functions and their derivatives by polynomials
along with Hermitian interpolation of some multiplicity at given points (see Section
2 below). Theorems in this topics can be applied, for example, to obtain precise
sufficient conditions for the possibility of expansion of a function in Fourier-Jacobi
series.
B . Functions are approximated by polynomials only in a one-sided way, i.e. from
above or from below. This topic was initiated by J. Karamata, and direct theorems on
one-sided integral approximation of smooth functions were obtained long ago by A. G.
Postnikov and G. Freud. Such theorems were used in the proofs of Tauberian theorems
with remainder terms (see Subhankulov [1], 1976, Ch.l). Comonotone approximations
are those where a function and its approximating polynomial are supposed to be
monotone, convex, etc. Pioneer works in this topics were done by G. G. Lorentz (see
Shevchuk [2], 1992 and Section 4 of this paper).
C . The coefficients of approximating polynomials are assumed to be integers. Then
certain arithmetical constraints are assumed to be true for the approximated function.
First sharp results on the order of approximation on [0,1], i.e. some analogs of the
35
36
R . M. T R I G U B
Jackson and Bernstein theorems, were obtained by Gelfond (1955, see [3]). Pointwise
approximations on an interval of the length at most four were studied by the author
[4] (1962), see Section 6 below.
Uniform approximation of functions by polynomials with positive coefficients are
studied in connection with the problem of the spectra of positive operators by Toland
[5](1996), see Section 5 below.
This paper attempts to review the mentioned constraints.
Below, с will denote different absolute, positive constants, j(a, b) are similar constants depending on a and b, Pn(W) is the set of algebraic polynomials, whose degrees
do not exceed n and whose coefficients belong to a given closed set W С C, P(W,, E)
the set of all functions admitting uniform approximation on E by polynomials with
coefficients from W. Further, we shall assume that R-
Z Ո R+ = -Z—
w(f; h)(wk(f;
= -R+
= (-<x>, 0], Z+ =
h)) is the modulus of continuity (of smoothness) with
the step h > 0 of a bounded function f : [a,b] ^ C (see, for example, Timan [6j, I960
or Dzyadyk [7], 1977), i.e.
Wk ( f , h ) =
sup
0<S<ha<x<b-rS
sup
E
( k К - 1 ) " f (x +
vS
v=0 ՝ • V
w = wi,
)
and for any x e [a, b] and n G N
Sn,a,b(x)
So,n,a,b(x)
=
1
v / ( x - a)(b
n
. J
1
n
= mm Հ - л/(x
n
- x) +
2n 2
T^
T 2 (x
- a)(b
- x),
Sn(x)
=
a)(b
;
b- a
Sn,-i,i(x),
-
l
x)
>.
2. A P P R O X I M A T I O N OF F U N C T I O N S B Y A L G E B R A I C
P O L Y N O M I A L S WITH H E R M I T I A N I N T E R P O L A T I O N
S. N. Bernstein payed attention to the fact that the sequence of best approximations
to a function by polynomials Pn (n = 1, 2 , 3 , . . . ) on the interval E n ( f ) of the real
axis decreases the condition Lip a inside the interval and Lip a/2 near the endpoints
1
of approximation by polynomials near the endpoints of the interval is almost twice
higher than that inside the interval. A direct theorem with modulus of continuity of
the r-th derivative was obtained by A. F. Timan (1951),fwhile the converse statement
was proved by V. K. Dzyadyk (1956) in the power case. A direct theorem containing
APPROXIMATION OF FUNCTIONS BY
POLYNOMIALS
37
modulus of smoothness uk of arbitrary order к was proved by Yu. A. Brudnyi (see,
e.g., Timan [6], 1960, Dzyadyk [7], 1977, Trigub-Belinsky [8], 2004).
The following statement is an improvement of the Timan-Brudnyi theorem, which
contains an additional estimate from below.
T h e o r e m 1. (2000, [9]) Let r e Z+. Then for any f e C r [-1,1] (к e N) and
n > max(k + r — 1, 3) there exists
Sn(xH
f
( r )
a polynomial
pn e Pn such that for any x e [-1,1]
; 5n(x)) < Pn(x) — f (x) < y(r,k)Sn(x)wk
f
( r )
; 5n(x))
This statement readily implies an approximative characteristics of the class of
functions f with f
( r ֊ 1 )
e Lip 1. However, the polynomial operator in Theorem 1,
unlike that in the periodic case (see, e.g., Dzyadyk or Trigub-Belinsky), is not linear.
Note that in the case of linear operators, the author earlier ([10], 1973) had to add
some boundary conditions. Anyway, some approximative characteristics of the same
r
The next theorem contains an asymptotically sharp result for the class
Wr
= { f : | f ( r - 1 ) ( x i ) — f(r-1)(x2)|
< |xi — x2|}
T h e o r e m 2. (1993, [11]) For each r e N there exists
some y = Y(r) such that for
any f e Wr [—1,1] an d n > r — 1 there is so me pn e Pn satisfying
for all x e [—1,1], where the
4
be replaced
by a smaller
inequality
constant
K r
cannot
the
:
~
( — 1)k(r +
П k-0 (2к +
1)
1)r+1
one.
Note that for r = 1 and K1 = n/2 the above inequality was proved by V. N.
Temlyakov [12] in 1981, and an earlier result of S. M. Nikolski contains the factor
ln n in the remainder. Besides, V. P. Motornyi [13] (1999) proved a similar result for
r
also similar results in the ^-metrics with the weight (1 — x 2 ) a (see [11], 1993), as
well as without a weight (see V. P. Motornyi and О. V. Motornaya [14], 1995). See
also Trigub-Belinsky [8].
38
R. M. TRIGUB
W
r
points, is proved by A. B. Tovstolis [15] (2001). Yet there is no such theorem for
functions in the exterior of the interval.
We turn to approximation with interpolation.
T h e o r e m 3. (2002, [16]) Let {x^^^
be different
points
in [-1,1]
and let
m
Xi(x)
= J J ( x - xթ),
So,n(x)
= m i n { S n ( x ) : \x - xk|, 1 < к < m}.
թ=1
Then for any r G Z + and k G N there is a constant
f G C r [-1,1
7 = 7(r, k, X1) such that for all
] and n > max{m(r + 1) - 1, к + r - 1} there exists
for which the
pn G P n
inequality
f ( x ) - pV (x) J < 7SO—
(x)wk
( f ( r ) ; S 1/ ( x ) S i
holds for any x G [ - 1 , 1 ] and 0 < v < r. In this inequality,
possible
a polynomial
degree of 50 n, while n is the least
- 1 / k
(x))
1/k is the
maximally
possible.
Gopengauz [17] (1967) and Kopotun [18] (1996) treated interpolation at the endpoints of the interval.
The constant 7 in Theorem 3 depends on the location of the interpolation points
խթ}, от more precisely on minM \xթ - x^+1 \. However, it turns out that sometimes
n
m
similar interpolating trigonometric polynomials, Fejer constructed some algebraic
polynomials of the degree 2m, while Bernstein decreased the order down to m(1 + e),
where e > 0 is arbitrary. The exact approximation order for Jackson interpolation
polynomials is found in [19] and [20].
T h e o r e m 4. (2006, [21]) For any system
N = N(e, {xs})
polynomial
\\f - TnII < (2 + e)El(f),
is the best approximation
than n satisfying
Tn(xs)
2- e
the
= f (xs)
in the C(T)-metrics
n
2+ e
and any e > 0, there exists
such that for each n > N and f G C(T) there is a
Tn of a degree not greater
where E' T(f)
{xs}m t
n
a
number
trigonometric
conditions
(1 < s < m),
by trigonometric
polyno-
APPROXIMATION OF FUNCTIONS BY
POLYNOMIALS
39
Note that a similar result is true for algebraic polynomials as well.
3. O N E - S I D E D AND C O M O N O T O N E A P P R O X I M A T I O N S
A result on the rate of one-sided approximation is contained in Theorem 1, where
f (x) < p n (x) for any x e [—1,1]). By adding some special polynomials to those of
the "good" approximation similar results can be obtained. For instance, Theorem 1
readily follows from the Timan-Brudnyi theorem and the statement c) from Theorem
5 with a = 0.
T h e o r e m 5. (2000, [9]) The following
a) Let an algebraic
polynomial
of some
are true.
qm(x) of the degree
[—1,1], and let all its zeroes
and the ratio
statements
lye outside
the ellipse
of axes X> 1. Then there exists
degree n < X m and a constant
max
m be positive
for any x e
with foci at + 1 and —1
a real algebraic
polynomial
y = Y(m) e (0,1), such
pn
that
1 — q m (x)pn(x) 1 < y-
1
xe[-1,1]
b) For any y e [—1,1] andn
such that
{\x — yl + Sn(y))
e N, there exists
a polynomialpn(x)
< pn(x) < c(lx — yl + Sn(y))
2
2
= pny (x) e Pn
—1 < x < 1,
,
where Sn(y) = n^/ 1 — y 2 + 4?.
c) For any a e R n e N and modulus
nomial
pn (x) such
Sn(x) a^k(Sn(x))
Hence,
of smoothness
uk, there exists
a poly-
that
< pn(x)
< Y(a, k)Sn(x) a^k(Sn(x)),
—1 < x < 1.
ppn? (1) > 0 for any v e Z + .
One can use the following representation for comonotone approximations of a
function f (x) satisfying the conditions f
^
^
r•
(1) = 0 0 < v < r:
с1
)r+1
f(x)
(v)
с1
f (y — x) rdf (r)(y)=
J
Jx
f
hry(x)df (r)(y),
—1
where
r,y ( x )
h
=
(x
y)
( )
2 r!
r
^n
( x
—
y)
— 1)
(r e
Z
+).
(x — y)
40
R . M. T R I G U B
T h e o r e m 6. (1999, [22]) For any y G ( - 1 , 1 ) , any r, s G Z + and any n > 2r + 1,
there exists
a polynomial
pn(x) = pny (x) of degree not greater
than n, satisfying
the
inequality
I sign (x - y) - pn(x)\
<
1 - x2
Yf
Sn(y)
1 - x 2 + 1 - y 2 + n 2 \signx - signy\/ \ J x - y\ + Sn(y) /
< 7(r, s)
for all x G [-1, 1]. The polynomial
of the derivative
positive
p'n(x)
or negative
pn(x) is increasing
in [-1,1]. Besides,
lie in [-1,1] and their leading
as an
coefficients
all zeroes
can be chosen
either-
option.
r=0
T h e o r e m 7. (2000, [9]) Given
p > 0, let E n ( f )
algebraic
not greater
polynomials
of degrees
the best approximation
by polynomials
p
be the best approximation
than n in Lp[-1,1],
with the constraint
while En,O(f)p
p— (1) > 0.
by
be
Then:
a) for any r G N and n > r + 1
SU
P
n,O
E
Ur+2(f;
,
(f
' ,
2/(r + 2)U
= sup
n,O
(f
E
= sup -
՝
*Er+1(fU
^s(f
s
u
p
E2(f
( r - 1
՝);2/3)c
(f)
En,r
=
(f ) oo
n,r
E
( r - 1 )
U
=
b) for any r G Z + p,q G (0, +ж) and n > r + 1
n,r
E
(f
P Ur+2(f (t ;2/(r
on
su
p
)
n,r
E
(f
I o^ = suP Er+1(f
J?
+ 2))q
=sup•
n,r
E
E1(f
p
)
(f
( r )
n,r
E
= suP —t*(
r)
)q
Ա2Ա
)p
)q
are taken over the functions
conditions
r+v
f
(1) = 0, and (-1) fdecreases
p
)
Л\
(r>;1)q
— ^
In a) and b), all upper bounds
(r)
(f
of C r [-1,1]
satisfying
the
on [-1, 1] for any v G [0,r],
A related result is applied to construction of some counterexamples to comonotone
approximation (Shvedov [23], 1980, also see Shevehuk [2], 1992). In the above form,
Theorem 7 can also be applied to approximation with interpolation at the endpoints,
to one-sided approximation, to approximation by positive operators or by polynomials
with positive coefficients, etc. (see below Theorems 12 and 14).
APPROXIMATION OF FUNCTIONS BY POLYNOMIALS 7
4. APPROXIMATION OF F U N C T I O N S B Y POLYNOMIALS
WITH C O N S T R A I N T S ON C O E F F I C I E N T S
In this section, we consider approximation of functions on sets of the complex
plane by polynomials pn with coefficients from a given set W (pn G P n (W)). This
inclusion means a restriction on the growth of either absolute values or arguments of
the coefficients.
Theorem 8. (1977, [24]) If
W7
= { ± | a n | : \an\ S TO, \an+1\-
then any real function
< A\an\ Y,
Ы
n >
of C[a, b] with 0 < a < b < 1 belongs
N},
to P(WY, [a, b]) where
Y < ln b/ ln a.
This theorem is an improvement of a result due to A. O. Gelfond (1965).
Let E be a compact set in С with connected complement and let A(E) denote
the set of all continuous functions on E, analytic at the interior points of E, if such
A(E)
the uniform limit of a sequence of algebraic polynomials. Which is the upper
bound
of the absolute
stated,
which conditions
values
of the coefficients
the sequence
{w(k)}™
f G A(E) and £ > 0 there exists
of all such polynomials?
satisfies
the polynomial
p such
max f ( z ) - Y , CkZ k <£,
zeE
if it depends
Otherwise
only on E and for any
that
\ck\< w(k),
к G Z+?
k=0
Theorem 9. (1977, [24])
I. If 0 G E, then for w(k) one can take R -k
dist (0, E), but not with R > d, generally
II. Let the origin
belong to the boundary
(k G Z+) with any R < d =
speaking.
dE of the set E. If it is not a limit
point
E
f G A(E) (f (0) = 0)
lim ( w ( k ) ) 1 / k = TO
k—
is a sufficient
values
condition
for the validity
of the above inequality
of the coe fficients of polynomials,
lim sup (w(k))
k—
and
1/k
зо
for the absolute
42
R . M. T R I G U B
is a necessary
condition.
Note that the general ease for 0 G dE is studied by V. A. Martirosian ([36], 1983).
We come back to approximations on an interval of the real axis, where there are
some constraints on the arguments of the coefficients.
L e m m a 1. (1977, [24]) If a G [0,1), then
ж
f ( x) = ^2 ckx k,
x G [a, 1],
k=0
ck > 0,
k G Z+,
is a necessary and sufficient condition under which a real function f G C[a, 1] belongs
to P(R+, [a, 1]).
J. F. Toland [5] (1996) considered approximation of functions by polynomials with
[a, 1]
a < 0
E
the origin.
Theorem 10. (1998, [25])
E
R and suppose
that E consists
finite set of segments
of the unit disk centered
lying outside
the disk. Further,
Then, f (z) = f (z) (z G E) and f (
sufficient
condition
k)
at the origin
let E Ո R+ = [0,1].
> 0 (k G Z+) presents
under which a function
and a
f G A(E) belongs
a, necessary
and
to P(R+, E).
II. Let W С С be a closed set such that XW С W for all X G R+. If the origin
E
lying on a ray starting
the condition
f (0)
(k)
E
at the origin,
then every function
G W (k G Z +) belongs
to
f G A(E)
satisfying
P(W,E).
In the next theorem, the polynomial coefficients can belong to two given rays
starting at the origin, an improvement of a theorem due to S. N. Mergelian.
Let E G С be a compart set with a connected complement and A(E) denote the
E
E
exist.
Theorem 11. (1998, [25]) Let E Ո (0, +TO) = $ and, Wa = R+ U e i a R + ,
a G (-n, 0) U (0, n).
where
APPROXIMATION OF FUNCTIONS BY
I. If0 G E, then A(E) = P(Wa,E).
f G P(R+,E)
43
If 0 G E, then every function f G A(E),
such that f (0) G Wa, belongs
II. Let, in addition,
POLYNOMIALS
P(Wa,E).
the compact
set E be symmetric
with respect
to R.
Then
if and only if f G A(E), f (z) = f (z) (z G E), along
with
f (0) G R+ if 0 G E.
Now, let us consider the problem on the approximation rate by polynomials with
positive coefficients under the additional restriction that approximations are either
one-sided (Theorems 12 and 13) or comonotone (Theorem 14).
T h e o r e m 12. (2001, [26]) Let r G Z+ and a function
the necessary
polynomial
condition
f
(v)
(0) > 0 for v G [0,r],
pn G P n ( R + ) such
be replaced
satisfy
Then, for any n > r there exists
a
that
0 < p+ (x) - f (x) < Y(r, a)6 rn,a,o(x)"
and и cannot
f G C r [a, 0] (a < 0)
( f ( r ) ; Sn,*,o(x))
,
by any иk with k > 1.
Denote polynomials with positive coefficients by p+ and polynomials with negative
pn T h e o r e m 13. (2001, [27]) Let r G N, and the total
f
(x)
on [a, 0] (a < 0) be less or equal
(r-1)
f
variation
of the
derivative
to 1. In case r > 1, we also
assume
(r-2)
[a, 0]
If in addition,
some
polynomials
[a, 0^, such
f (v)(0)
= 0 for each v G [0, r - 1], then for any n G N there
p+ and p- satisfying
that
f p+ (x) J a \/\x\(x
the inequalities
p - ( x )
- a)
exist
p—(x) < f (x) < p+(x) on
dx < Y ( r ^ a ) 1 .
In the above theorem, the polynomials p+(f)
nr
and p - ( f ) present linear operators,
and their coefficients cannot be of opposite signs.
T h e o r e m 14. (2001, [27]) If a function
f(0)
polynomial
> 0
p+ such
n > 2
f G C[a, 0] (a < 0) is increasing
[a, 0]
that
If (x) - p+(x)I
< Y ( a ) J f ;
.
(decreasing)
44
R . M. T R I G U B
The statement of Theorem 14 with
instead of ш and no assumptions on the
coefficients was proved by DeVore and Yu [28] (1985). In Theorem 14, ш cannot be
replaced by ш2.
5. I N T E G E R C O E F F I C I E N T S
The possibility of approximation of functions by polynomials with integer coefficients was studied by Fekete, Szego, Hewitt and Zukerman and Alper, while theorems
on the approximation rate of smooth functions were obtained by A. O. Gelfond
and by the author. In these problems, the assumption that the polynomials possess
integer coefficients modul less than one is essential. Besides, a necessary and sufficient
condition for the existence of such polynomials is that the transfinite diameter of the
set is less than one (if the considered set is an interval, then its transfinite diameter is
equal to the quarter of its length). Zeros of such polynomials dictate certain arithmetic
conditions on the function. For instance, if 0 G E, then f (0) G Z. D. Hilbert used
Minkowski's theorem on linear forms to prove the existence of such polynomials, as
well as to estimate the maximum of their absolute values. Later, B. S. Kashin [29]
(1991) improved these estimates in a different way. S. N. Bernstein posed the question
on the approximation rate of a constant
by polynomials with integer coefficients (see
the surveys: Trigub [30], 1971 and Ferguson [31], 1980, see also Montgomery [32],
1994).
Below, we give some direct theorems in a stronger form, with some additional
restrictions.
Theorem 15. (2001, [27]) Let for some
continuous
and \f (x)!
arithmetic
conditions
satisfied.
< 1 almost
(r)
f
(v)
r G N the derivative
everywhere
(0)/v! G Z and f
(v)
in [0,1]. Besides,
f
be
absolutely
let the
necessary
( r - 1 )
(1)/v! G Z for any 0 < v < r — 1 be
Then for any n > 4r + 2 there exists a polynomial
qn G Pn (Z) such that for
any x G [0,1]
Theorem 16. (2003, [33]) Let [a, b] С R and {xk}m=1
in [a, b] along
with their algebraic
conjugates.
be all integral
algebraic
Let X be a polynomial
with
numbers
integer
APPROXIMATION OF FUNCTIONS BY POLYNOMIALS
coefficients
and leading
1 < k < m) in [a,b].
coefficient
Besides,
1, such
that \X(x)\
45
< 1 and X(x) = 0 (x = xk,
denote
X i ( x ) = J J ( (xx - xk).
k=i
Then the following
statements
are true:
a) If f G C r [a, b] for some
tion polynomial
hf of f , defined
hfv)(xk)
are integers
r G Z+ and all coefficients
= f (v)(xk),
(this condition
there is a polynomial
by the
interpola-
equalities
1 < k < m,
0 < v < r,
is also necessary),
gn with integer
of the Hermite
then for any n > (r + 1)m — 1
coefficients
and so me degree < n, such
that for any x G [a, b] and v G [0, r]
f
(v)
(x) — g n v ) (x)| < Y(r,X)ՏՈ;Հե(x)
where
Sn,a,b(x)
b) I f , in addition,
can be replaced
խ f
2
= 7 ֊
^~\/(x
(b — a)a)nn
( r )
; Sn,a,b(x))+
1
— a)(b — x) + ֊ 22 .
n
X(a) = X(b) = 0 and all zeroes of X He in [a, b], then Sn a b(x)
by
min { S n a b b ( x ) : \x — xk \(b — a) -m,
and the second
removed.
summand
in the right-hand
This is approximation
f
decreases,
(v!) f (
along with Hermite
—1) G Z and ( v ! ) f (0)
-1 (v)
- 1 (v)
for each n > 2r + 2 there exists
1< k<
side of the inequality
T h e o r e m 17. (2003, [33]) Let r be an odd number.
( r )
Ka,b(.x)
a polynomial
in a) can be
interpolation.
If f G C r [—1, 0], the
derivative
G Z+ for any v G [0,r], then
qn with natural
coefficients,
such
that
f (x) < q n (x) when x G [—1, 0], and for any v G [0,r]
f
(v)
( x ) — qn v ) (x)| <
(x) խ ( f ( r ) ; Sc,n(x^ + So,n(x)
where
So,n(x)
Above,
either
even one.
= min <
[—1,0] can be replaced
^\x\(x_
n
+ 1)
, \x\(x + 1 H .
by [—1 — e, 0] (e > 0) or the odd, number
r by an
46
R . M. T R I G U B
N
interval wider than [-1,0] can be considered.
The proof of the next lemma is based on the extremal properties of the Chebyshev
polynomials and certain arithmetic properties of their coefficients.
L e m m a 2. (2003, [33]) The following
a) For any numbers
natural
g(m+1)
2
are true:
r,m,n G N, there exist two polynomials:
coefficients,
2r
statements
q1n and g2n
with
such that for any x G [-2, 0]
x r + x r+ mq1n(x)
<
Y(r,m) min < \ x \ , ^
<
Y(r,m) min
and
2 r ՜m(^m+1)
֊ 1 xr
r+m
q2,n(x)
n GN
\x\,^
r
r
qn
x G [-1, 0]
1
0 < -x - x qn(x) < cmin լ \ x ( x + 1)\, — j .
Now, we turn to the constant approximating problem.
Theorem 18. (1962, [4]) If p G [1, ж), then:
a) En (X; [0, b], Z)p x n -2/p
b) If an interval
for any b G (0,1] and A G (0, 1).
(a, b) with b - a < 4 contains
En(A;
[a, b],
Z)p x
a,է least one integer,
then
n -1/p
for any A G (0,1).
The next theorem shows the cases when the decreasing order of best approximations
En(A;
[а, в], Z+) of a const ant A G (0,1) by polynom ials q+ G P (Z+)
can be found.
Theorem 19. (2001, [27]) If -1 < а < в < 0 and A G (0, 1), then the
statements
are true:
I. If \в\ < \а\(
— ) or а + в =
1 - а
V—+M(VW\
+ V—)
q
for som e q G N and
<
2(V—\-VW\
then
En(A;
[а, в], Z+
v l — ^ в Т
vl^
+y/W\,
n
following
APPROXIMATION OF FUNCTIONS BY
where the sign x means
a two-sided
inequality
POLYNOMIALS
with positive
47
constants
indepen-
dent of n.
II. If \в\ > \a\( —+— )
V1 - a z
numbers
and a = —, then for any X = -P- with any
q
qs
p, q, s and for infinitely
En(X;
many
values
of n
[a, в], Z+) x \a\ n,
n
1
p
III. If0,1 < \в\ < — < \a\ < 1 + в and X = —,
then for infinitely
2
qs
n
En(X;
natural
[a, в], Z+) x
p
2
IV. If X = —,
a + в = — and в =
qs
q
of n
many values of
2 -n,
n
r
т, wherep,
rq + 1
q, s G N, then for all values
n
M X; [a,i3 ], Z+)
V. For any r G (0, 1) and a G (0,n/2)
K = Kr
and suppose
a
we write
= {z G C : \ z \ < r, Re z < —r cos a }
that f (z) = X G R \ Z.
lim Ei /n(X;
n^<x>
Then
K, Z+) = 2 s i n
^
)
2(2n — a)
na
na
for
anyУ r < 2 sin —
. Besides,
if r > 2 sin —
J
՜
2(2n — a)
' J
2(2n — a)
2շ) and X is not dyadic-rational,
then
lim sup E lJ n{X;
n
—
(for instance
u
r =
K, Z+) = 2 ֊ .
2
So far, there are no theorems on approximation of analytic functions on the general
sets of the complex plane, the case of the unit square is considered by Vit.Volchkov
[34] (1996).
6. P O I N T W I S E A P P R O X I M A T I O N OF P E R I O D I C F U N C T I O N S B Y
T R I G O N O M E T R I C P O L Y N O M I A L S WITH H E R M I T I A N
INTERPOLATION. PIECEWISE ONE-SIDED APPROXIMATION
All well-known theorems on the approximation rate of smooth functions are based
on the modulus of smoothness
of a smooth function and its derivatives (see,
48
R . M. T R I G U B
eg., Dzyadyk [7] and Timan [6]). In the next theorem, the Hermitian interpolation
constraint is added, as well as the location of the point is taken into account.
We suppose that T = [—n,n],
from C r(R),
||g|| = sup \g(x)\
C r(T)(r
e Z+) is the set of 2n-periodic functions
n
and тп(x) =
Y
cke ikx
k=-n
n
that x i < X2 < • • • < xm < x i + 2n =
x
m +
i.
T h e o r e m 20. (2006, [21]) For any function
of interpolation
polynomial
points
f e C r(T)
(r e Z+), any
m(r+1)
2
խտ}™, any к e N and any n >
тп of degree not exceeding
f (v)(x)
is a trigonometric polynomial
- Tn v)(x)\<
there
system,
exists
a
n, such that for any v e [0, r] and x e R
Y(r,k, { x s } ) S n -
f
v
( r )
1
; 1(nSn(x)) 1/^
,
where
= min <{
Sn(x)
The degree of nSn(x)
cannot
: sin
1
n
be greater
1 < s < m >.
2
than
1/k.
Now, let us find an explicit form for the trigonometric Taylor polynomial, i.e. the
polynomial of the least degree that satisfies т (v)(0)
= f (v)(0)
(0 < v < r). First,
suppose that r = 2n, i.e. is an even number.
Naturally, the considered polynomial must be of the form
2n
£ f
(k)
(0)^k,n(x),
k=0
where hk,n is a polynomial of degree not exceeding n, while hk,n(x)
O(x 2n+1)
as x ^ 0. Besides,
2
= 1,
ho,n
n
= ^ т г ( 1 - c ° s x ) n.
(2n)!
հշո,ո
k,n
h
L e m m a 3. (2006, [21])
n
n
) Ц Qn(x) = J J ( x + k 2 ) = ^^
a
k=1
then
&s,nx s,
s=0
n
h
k,n
( x )
=
Y
s=[ Ч
a
1
]
s , n
h
2 n -
k ) ( x )
.
= x k/к!
+
APPROXIMATION OF FUNCTIONS BY
b) If к e [0,n],
then
(x)=y
՚
=
n
k n
49
= h'2k n and
եշս֊ւ,ո
4_k2֊s
n
h2k2
POLYNOMIALS
bss kkV (1 — cos x) s,
'
d's
(
bs sk k = , , ՝ . , — г т 0 2ss (arcsin
x) 2k
'
(2k)\(2s)\dx \ K
'
The statement a) of the above theorem is obtained by the author and V. P.
Zastavnyi.
If r = 2n — 1, i.e. is an odd number, է hen can hk, n be replaced by h'k+1 n, and the
following theorem is true.
T h e o r e m 21. (2006, [21])
a) For any function
9 e [0,1] such
f e C 2n(T)
and any point x e — n, Щ] ? there exists
some
that
2n
f (x) — J 2 f (k)(0)hk,
n(x) = [D2nf
(9x)
— D2nf
(0)] h2n,
n(x),
k=0
where D2nf
(x) = qn(d 2/dx 2)f
b) For any function
(x) and qn is as in Lemma
f e C r (T) (r e N), there exists
3.
a trigonometric
polynomial
ե f , such that its degree does not exceed, [ +r] and for any v e [0, r\ an d, x e R
r
f ? (v(v)\x)
(x)
v) v)
—
— hfhf(x)\<
where и is the continuity
Y(r)u(f (r);
modulus
П
x
r ֊ v
2
sin
of f .
Piecewise one-sided approximations are not studied yet, though such an approximation implies, for example, copositive approximation since the sign of the approximating polynomial remains the same as that of the function on a finite number of given
intervals (the function f is real and the sign of f — Tn alternates). It is obvious that
the number m of the points of interpolation has to be even since the function and
the polynomial are periodic. As in the case of one-sided approximation, some special
polynomials should be constructed using algebraic polynomials (see Section 3 above).
L e m m a 4. (2006, [21]) There exists
an absolute
N an odd, trigonometric
Tn of degree not exceeding
polynomial
constant
c> 1 such that for any n e
such that in [0, n]
and hence Tn(n — x) =
1 + n sin x
Tn(x).
< Tn(x)
< cn
1 + n sin x'
n can be
constructed,
50
R. M. T R I G U B
Theorem 22. (2006, [21]) Let f e C 1 (T) (k e N) and let
be some interpolation
degree not exceeding
points.
{x.s}T,
Then for any n > m/2 there exists
n, such that for any x e [xs,xs+1]
" Հ f ' ; 1 ) Sn(x) < (-1) s[f
(x) - тп(x)]
where m is even,
a polynomial
тп of
(s e [1, m])
< Y(k,
f'; ^
Sn(x).
The proofs of theorems 4,20-22 and lemmas 3-4 see [35]. A similar statement is
true for approximation by algebraic polynomials on an interval (see Theorem 1).
Finally, we mention that in the periodic case the condition about positivity of the
coefficients can be added only for a special class of functions. Besides, if
lim
n—
П
k,ne ikt dt = 0,
f (t) - Y
П
k=
c
n
then
lim
lim \ck,n\ = 0.
kl^oo n—ж,
f
L by polynomials satisfying the condition i n f k n \ck,n\ > S > 0.
If the same convergence condition is valid in L and ck,n > 0 for all k and n, then
f
of the interval (-a, a), where a e
(0,n).
Removing an arbitrarily small neighborhood of zero makes approximation by polynomials with positive coefficients possible.
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Поступила 12 февраля 2009